Simplexes

The following assertions hold: If P is the unit n-hypercube, then the union of the n-simplexes formed by the convex hull of each n-path is P, and these simplexes are congruent and pairwise non-overlapping.

If some of the simplexes occur in the set more than once, these are prefixed with an integer count.

That explains why the simplexes are non-overlapping.

The fact that the union of the simplexes is the whole unit n-hypercube follows as well, replacing the strict inequalities above by " ".

The same arguments are also valid for a general parallelotope, except the isometry between the simplexes.